PHYSICAL REVIEW RESEARCH 2, 013312 (2020)Editors’ Suggestion

Vector-borne epidemics driven by human mobility

David Soriano-Paños ,1,2 Juddy Heliana Arias-Castro,3 Adriana Reyna-Lara ,1,2 Hector J. Martínez ,3

Sandro Meloni ,4 and Jesús Gómez-Gardeñes 1,2,5

1GOTHAM laboratory, Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, 50018 Zaragoza, Spain2Departamento de Física de la Materia Condensada, Universidad de Zaragoza, 50009 Zaragoza, Spain

3Department of Mathematics, Universidad del Valle, 760032 Santiago de Cali, Colombia4IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), 07122 Palma de Mallorca, Spain

5Center for Computational Social Science (CCSS), Kobe University, 657-8501 Kobe, Japan

(Received 16 December 2019; accepted 24 February 2020; published 13 March 2020)

Vector-borne epidemics are progressively becoming a global burden, especially those related to flaviviruses,and the effects of different factors such as climate change or the increase of human mobility can sensibly increasethe population at risk worldwide. Such outbreaks are the result of the combination of different factors includingcrossed contagions between humans and vectors, their demographic distribution and human mobility amongothers. The current availability of information about all those ingredients demands their incorporation intocurrent mathematical models for vector-borne disease transmission. Here, relying on a Markovian formulationof the metapopulation dynamics, we propose a framework that explicitly includes human-vector interactions,mobility, and demography. The analysis of the framework allows us not only to derive an expression of theepidemic threshold capturing the conditions for the onset of the epidemics but also to highlight some unseenfeatures of vector-borne epidemics, such as abrupt changes in the unfolding patterns of the disease for smallvariations of the degree of mobility. Finally, driven by these insights, we obtain a prevalence indicator to rankpopulations according to their risk of being affected by a vector-borne disease. We illustrate the utility of thisindicator by reproducing the spatial distribution Dengue cases reported in the city of Santiago de Cali (Colombia)from 2015 to 2016.

DOI: 10.1103/PhysRevResearch.2.013312

I. INTRODUCTION

The explosive dissemination of Zika virus across the Amer-icas has been one of the major concerns of public healthorganizations across the world in recent years [1]. Zika’sglobal threat is, unfortunately, the last example of the ex-tremely rapid dissemination of mosquito-borne flavivirusesover the past two decades. From Dengue to Zika, throughWest Nile and Chikungunya viruses, more than one billionpeople are infected and more than one million people diefrom vector-borne diseases (VBD) every year [2]. Accordingto the World Health Organization (WHO), VBD are respon-sible for one-sixth of the illness worldwide and more thanhalf of the human population live in risk areas for thesediseases [3].

The threat of emergent VBD in tropical and equatorialregions progressively span across more temperate areas as abyproduct of climate change. As temperature rises, the areasthat are conducive to mosquitoes expand, meaning more op-portunities for VBD to spread [4–9]. Another important driverof many recent VBD outbreaks is the explosion of human

Published by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.

mobility we have witnessed over the last decades, includingurban, regional and long-range displacements [10–14]. Inspite of the social and economic benefits, the expansion ofhuman mobility networks produced, as a consequence, thespeed-up of epidemic waves and the emergence of correlatedoutbreaks in faraway regions.

When facing these VBD outbreaks, public health systemshave to respond quickly and efficiently to mitigate the spreadof the disease. Although vaccines are a long-term solution, upto now, the ones available for some flaviviruses have limitedimpact [15,16]. Thus the most common way for preventinglocal Aedes-VBD outbreaks is tackling the vectors by the useof pesticides, larvivorous fishes, or Wolbachia bacteria [17].The use of geolocalized control strategies, however, may notalways be as effective when facing the threat of a global-scale pandemic [18]. On the contrary, the fast transcontinentalmovement of VBD demands a coordinated action of all theinvolved actors for the efficient use of local control means.This implies taking into account that those populations atrisk are not isolated and, as for human-human transmissiondiseases [19], human mobility plays a key role in the spreadof VBD across different populations [20]. Therefore, incor-porating human mobility into disease transmission modelshas become a must when proposing mathematical frameworksaimed at capturing the contagions patterns observed in actualepidemic scenarios. So, the creation of new policies could relyon those refined models to generate better strategies involvinghuman mobility to contain epidemics spreading.

2643-1564/2020/2(1)/013312(12) 013312-1 Published by the American Physical Society

DAVID SORIANO-PAÑOS et al. PHYSICAL REVIEW RESEARCH 2, 013312 (2020)

Metapopulation models, originally proposed in the fieldof ecology [21–23], enable the mixing of mobility and con-tagion dynamics into a single formulation. These modelscan be described as networks in which nodes account forgeographic locations (such as neighborhoods in cities, citieswithin countries, etc.), i.e., subpopulations where large collec-tivities live and interact. In addition, the links of the networkrepresent (and quantify) trips made by individuals betweendifferent subpopulations. From the first studies making useof metapopulations for the study of infectious diseases trans-mission [24–29], the field has advanced both in its theoret-ical grounds [30–35] and its use for large-scale agent-basedsimulations [36–38]. The latter approach incorporates manyrealistic aspects of human interactions with the goal of beinguseful for making epidemic forecast and the design of efficientprevention policies. On the other hand, the theoretical parthas been spurred by the increasing spatiotemporal resolutionof current data gathering techniques and many efforts havebeen devoted to bridge the gap between theory and realisticmodels during the last years [39]. First attempts in this direc-tion involved displacement kernels [40,41] to model a localrange of movement around an area that, in the last years,led the way to the inclusion of more sophisticated mobilitypatterns, such as the commuting nature of human mobility[42–46], the high order memory of human displacements [47],or the coexistence of different transportation behaviors [48].These sophisticated theories allow to capture the temporaland geographical spread of diseases while providing insightsabout the mechanisms driving the observed patterns.

In the case of VBD, the use of metapopulation modelshas been recently fostered due to recent outbreaks such asZika and Chikungunya. This way, epidemic models for VBDtransmission have abandoned mean-field and well mixed hy-pothesis to consider patchy environments subjected to hu-man flows. On the theoretical side, and pretty much as formetapopulations models of human-human transmission dis-eases, the frameworks rely on important assumptions thatallow its analytical study. One of these assumptions is toconsider the random diffusion of humans across patches[49–51] or displacement kernels [52] instead of realistic mo-bility patterns. On the other hand, when actual ingredientsof human mobility, such as its recurrent nature, are takeninto account both random diffusion [53,54] and displacementkernels models [55] fail to provide insights about the rolethat real mobility networks play on the transmission of VBD.Thus metapopulation theories are still far from incorporatingthe many aspects influencing the onset of VBD outbreaks andlack the predictive power provided by data-driven agent-basedsimulations [56,57].

The main goal of this work is to provide a benchmarkthat allows the study of large-scale vector-borne epidemicsin a unified way and, more importantly, enabling the test ofcoordinated control strategies at the light of available dataabout vector incidence together with human demography andmobility datasets. To this aim, we first elaborate a metapop-ulation model for the transmission of VBD that allows us toderive the conditions under which epidemics take place. Theanalytical expression of the epidemic threshold is revealedby a matrix encoding the probability of crossed infectionsbetween humans and vectors of different subpopulations.

Importantly, the spectral analysis of this matrix reveals the riskassociated with each patch, pointing out those subpopulationstriggering the epidemic onset. We confirm these results bytesting synthetic metapopulations and a real case, the cityof Cali (Colombia). To round off, taking advantage of thetheoretical insights and analytical expressions provided by theformalism, we propose a metric capturing the risk associatedwith each patch. We implement this metric in the city ofCali obtaining a very good agreement between the estimatedrisk and the actual distribution of Dengue incidence acrossdistricts, highlighting the important role of recurrent humanmobility patterns for explaining the spatial dissemination ofVBD.

II. METAPOPULATION MODEL FORVBD TRANSMISSION

In the following, we focus on the description of a vector-borne contagion dynamics in a complex metapopulation. Tothis aim, we consider a set of N populations or patchesin which contagion processes occur governed by the Ross-Macdonald (RM) model. The RM model [58–60] assumesthat both vectors and individuals can be either susceptibleof contracting the disease or infectious entities capable offostering the disease. In this sense, this model was one of thefirst attempts to address the impact of human-vector crossedcontagions on the spread of diseases. However, let us remarkthat, the original RM model has some limitations such asassuming that VBD do not confer long-lasting immunity toinfected individuals after overcoming the disease or includinga closed population of both humans and vectors. Nevertheless,despite these assumptions, it is able to capture the essentialingredients involved in the transmission of many VBD that donot confer immunity after infection, e.g., Malaria, or those dis-eases conferring immunity but having several co-circulatingserotypes within the same population, e.g., Dengue.

Additionally, here, we mainly focus on VBD provoked byflaviviruses, in which humans act as reservoir of the diseaseand mosquitoes are the vectors intermediating its transmissionamong individuals. We choose the original formulation of theRM model since it constitutes the minimal model of crossedinfections enabling the analytical study of the interplay be-tween human mobility and the spread of VBD. However,different disease-specific contagion patterns, such as human-human contagions in Zika [61,62], different epidemiologicalfeatures such as passive immunity [63], or the effects of vol-untary vaccination [64,65] can still be easily accommodatedby introducing small refinements to our original framework.

The relevant variables of the RM dynamics are (i) thefraction of infected humans at time t , ρH (t ), and (ii) thefraction of vectors infected at time t , ρM (t ). The evolutionof these two variables is given as a product of the elementaryprocesses described in Fig. 1(a). Namely, susceptible humansbecome infected with probability λMH after being bitten byan infected vector whereas healthy vectors become infectiouswith probability λMH when interacting with an infected hu-man. In addition, we assume that each vector makes a numberof β contacts with (healthy or infected) humans. This way, nohuman-human or vector-vector direct infections are allowed.Finally, infected humans become susceptible with probability

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FIG. 1. Ross-Macdonald model and metapopulation approach. (a) Schematic representation of the processes described in the RM model.Relevant parameters are (i) the probability that an infected vector transmits the disease to a healthy individual, λMH ; (ii) the probability that aninfected human transmits the disease to a healthy vector, λHM ; (iii) the feeding rate of vectors β; (iv) the probability that an infected humanrecovers, μH ; and (v) the mortality rate of vectors, μH . In (b) and (c), we show schematically the two basic stages of each Monte Carlostep in our metapopulation approach. As shown in (b), individuals are associated with one of the 3 nodes of a network. Namely, startingfrom the subpopulation in the top (1) and following clockwise we have populations composed of 8, 10, and 4 humans with 5, 3, and 9vectors respectively. Once the movement has been done [see (c)] individuals mix and, consequently, the instant populations of humans at eachnode change to 6, 8, and 8. The RM dynamics then takes place among the individuals and vectors coexisting at that moment in the samesubpopulation. Finally, after these interactions, the individuals go back to their respective associated nodes and the configuration is again theone of (b). Note that we consider that vectors do not move from their corresponding node.

μH , while (healthy or infected) vectors die with probabilityμM , being replaced by newborn healthy ones.

Although the RM dynamics captures the elementary con-tagion processes taking place inside each population, thedynamical evolution of each patch depends strongly on theothers, since they are not isolated. On the contrary, manyindividuals with residence in one subpopulation may visitothers during, for instance, their daily commuts to othergeographical locations. On the other hand, the mobility ofvectors is rather limited, hence they are assumed not to movefrom their original population. This assumption is valid formany VBD such as Dengue, Zika, or Chikungunya since theircarriers, Aedes mosquitoes, typically fly an average of 400meters [66]. Thus it is the mobility of infectious individuals(who pass the diseases to healthy vectors living in distantsubpopulations) what triggers the propagation of local diseaseoutbreaks across the whole system.

To characterize the mobility of individuals, we denote eacharea as a patch in a metapopulation. Each patch i has apopulation of ni individuals and mi vectors and, importantly,they may be different from one population to the other,as they are derived from the demographic partition of thepopulation and the observed vector prevalence in each patch.Each individual is associated to one subpopulation, say i,considered as her residence. This way, the population ni of thepatch i is the number of individuals whose residence is node i.Simultaneously, nodes are connected in pairs forming a com-plex weighted and directed network encoded in an adjacencymatrix R, whose entries Ri j account for the probability thata trip departing from patch i has as destination population j[see Fig. 1(b)]. Matrix R can be computed from the observed

number of trips between each pair of nodes (i, j), Wi j , as

Ri j = Wi j∑Nl=1 Wil

. (1)

Thus matrix R encodes the information provided by mobilitydatasets.

The former two dynamical processes at work—RM dy-namics and human movements—interplay in each time step ofthe metapopulation dynamics as follows. We start with a smallfraction of infected humans and/or vectors. The initial quan-tity, being small, can be homogeneously distributed across thepopulations or localized in one or few nodes in case of beinginterested in determining those patches boosting epidemicspreading in its early stage. Once the initial infectious seedhas been placed, the following microscopic processes areconsidered.

(1) At each time step, t , healthy agents decide to movefrom their residence with probability p or remaining in itwith probability (1 − p). Moreover, as symptoms associatedto some VBD are severe, we also include the possibility ofrescaling the infected agent mobility to αp with α ∈ [0, 1].

(2) If an agent leaves her residence, say i, she goes to adifferent subpopulation chosen among those connected to i.The choice is dictated by matrix R in Eq. (1), being Ri j theprobability of moving from i to subpopulation j.

(3) Once all the individuals have been placed in the patches[see Fig. 1(c)], humans and vectors that are currently atthe same patch interact as dictated by the RM model. Thisway, both humans and vector update their dynamical states[susceptible or infected as shown in Fig. 1(a)] as a result ofthe contagion and recovery processes.

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(4) Once the epidemic state of the agents have been up-dated, each individual moves back to her residence and theprocess starts again for time t + 1.

III. MODEL EQUATIONS

Once defined the basic steps of the mechanistic simula-tions, we now tackle the mathematical formulation of theprocesses described above. To this aim, for each patch i(i = 1, . . . , N) we have two variables: the probabilities thathumans with residence in i, ρH

i (t ), and vectors associated toi, ρM

i (t ), are infectious at time t , respectively. These 2N vari-ables evolve according to the following Markovian equations:

ρHi (t + 1) = ρH

i (t )(1 − μH ) + (1 − ρH

i (t ))IHi (t ), (2)

ρMi (t + 1) = ρM

i (t )(1 − μM ) + (1 − ρM

i (t ))IMi (t ), (3)

where IHi (t ) and IM

i (t ) account for the probability that ahealthy human with residence in subpopulation i and a healthyvector associated to i are infected at time t respectively. Theformer infection probability reads

IHi (t ) = (1 − p)PH

i (t ) + pN∑

j=1

Ri jPHj (t ) , (4)

where PHi (t ) is the probability that an agent placed in popula-

tion i at time t is infected. This probability can be written as

PHi (t ) = 1 −

(1 − λMHρM

i

1

neffi (ρH (t ), α, p)

)βmi

. (5)

Finally, neffi (ρh(t ), α, p), which is the number of humans

placed in (but not necessarily residing in) population i can beexpressed as:

neffi (ρH (t ), α, p) = [

1 − p(1 − (1 − α)ρH

i (t ))]

ni

+ pN∑

j=1

Rji(1 − (1 − α)ρH

j (t ))n j . (6)

In the same fashion, the expression for IMi (t ) in Eq. (3) reads

IMi (t ) = 1 −

(1 − λHM ieff

i (t )

neffi

)β

(7)

where ieffi (t ) is the number of infected humans placed in

population i at time t

ieffi (t ) = (1 − αp)niρ

Hi (t ) + αp

N∑j=1

Rjin jρHj (t ) . (8)

The above equations describe the time evolution for theVBD incidence, �ρH (t ) = {ρH

i (t )} and �ρM (t ) = {ρMi (t )}, in a

collection of connected patches with arbitrary demographic,�n, and vector, �m, distribution. Thus, by iterating Eqs. (2) and(3) starting from a given initial condition �ρH (0) and �ρM (0),we can monitor the spatiotemporal propagation of VBD andevaluate the steady epidemic prevalence on each geographicalarea.

Let us first study the steady epidemic prevalence acrosspatches �ρH as a function of the contagion probabilitiesλMH , λHM , and the degree of human mobility p in thepopulation. To reduce the number of parameters and without

loss of generality, let us define λHM = λMH = λ. We startanalyzing the case in which human mobility is governed byan unweighted undirected Barabási-Albert network (BA) [67]of N = 50 patches and average degree 〈k〉 = 4, all of themhomogeneously populated by ni = 1000 agents. Concerningvector distribution, we consider that the ratio between vectorsand humans populations inside a patch i, denoted in the fol-lowing as γi, is randomly drawn from a uniform distributionwithin the range γi ∈ [0.3, 1.7]. The top panels in Fig. 2reveals the great agreement between the predictions obtainedby iterating Eqs. (2)and (3) and the results obtained frommechanistic simulations for both the cases in which infectedagents mobility is totally restrained, i.e., α = 0 [Fig. 2(a)], andwhen there is no influence of the disease on agent mobility,α = 1 [Fig. 2(b)].

Finally, we check if the formalism is able to capture thespatiotemporal unfolding of VBD. For this purpose, we startby setting a seed localized in a single patch of the BA network,and then monitor the temporal evolution of the fraction ofthe population affected by the disease inside each area. Thebottom panels of Fig. 2 show the evolution obtained by iterat-ing Eqs. (2) and (3) [Fig. 2(c)] and by tracking the individualstate of each agent in Monte Carlo simulations [Fig. 2(d)].Again, it becomes clear that the formalism also reproducesthe different propagation pathways of VBD in the syntheticmetapopulation, despite the noise induced by the stochasticnature of the mechanistic simulations.

IV. ESTIMATION OF THE EPIDEMIC THRESHOLD

The validation of the Markovian formalism offers thepossibility of saving computational costs by iterating 2 × Nequations instead of performing lengthy agent-based simula-tions. However, the advance behind Eqs. (2)–(8) is that theyalso allow for deriving metrics and analytical results aboutthe dynamical behavior of VBD in large metapopulations. Arelevant quantity that can be analyzed is the epidemic thresh-old, i.e., those conditions that turns the epidemic state into astable solution of Eqs. (2)–(8). To address it, let us assumethat the disease has reached the stationary state so ρH

i (t +1) = ρH

i (t ) = ρHi , ρM

i (t + 1) = ρMi (t ) = ρM

i ∀i. In this case,Eqs. (2) and (3) turn into

μHρHi = (

1 − ρHi

)IHi , (9)

μMρMi = (

1 − ρMi

)IMi . (10)

As we are interested in computing the epidemic threshold,we now assume that the impact of the disease is finite butnegligible for all geographical areas, which in mathematicalterms implies that ρH

i = εHi � 1, ρM

i = εMi � 1 ∀i. This as-

sumption allows us to linearize Eqs. (9) and (10) yielding:

εHi =

N∑j=1

λMHβ

μH

(pRi j

mj

neffj

+ (1 − p)δi jmi

neffi

)︸ ︷︷ ︸

Mi j

εMj , (11)

εMi =

N∑j=1

λHMβ

μM

(αpRji

n j

neffi

+ (1 − αp)δi jni

neffi

)︸ ︷︷ ︸

Mi j

εHj , (12)

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VECTOR-BORNE EPIDEMICS DRIVEN BY HUMAN … PHYSICAL REVIEW RESEARCH 2, 013312 (2020)

0 50 100 150 200 250

Time (days)

0 50 100 150 200 250

Time (days)

1

10

20

30

40

50

Pat

ch

ρH

λλ0.4

0.3

0.2

0.1

0

50

30

40

20

10

1250200150100500 250200150100500

(a) (b)

(c) (d)

Time Time

0

.05

0.1

.15

0.2

.25 1

0.8

0.6

0.4

0.2

00.1 0.2 0.3 0.40.1 0.2 0.3 0.4

0.25

0.20

0.15

0.10

0.05

0

FIG. 2. Validation of the Markovian formalism. (Top) Epidemic size ρH as a function of the contagion rate between vectors and humansλHM = λMH = λ and the human mobility p (color code). Dots correspond to results of averaging 50 stochastic realizations whereas solid linesrepresent the theoretical predictions obtained by iterating the equations of the formalism. Model parameters have been fixed to (β,μH , μM ) =(1, 0.3, 0.3). The values assumed for the restriction of the mobility of infected agents are (a) α = 0 and (b) α = 1. (Bottom) Temporal evolutionof the fraction of infected agents (color code) inside each patch obtained by iterating Eqs. (2)–(8) (c) and by tracking the individual state ofeach agent while performing a single Monte Carlo simulation (d). Infectivity is fixed to λ = 0.3 whereas mobility parameter p is set to p = 0.3.The rest of parameters are the same as in (a).

where neffi has been defined as neff

i = neffi (0, α, p). Note that

the form of the elements of these two matrices depends onboth the mobility properties (p, R) and the demographicdistribution of both agents and vectors (�n, �m). For the sake ofclarity, let us write the former system of equations in a morecompact way:(

�ε H

�ε M

)=

(0 βλMH

μH MβλHM

μM M 0

)(�ε H

�ε M

). (13)

Equation (13) makes evident the bipartite nature of theprocesses involved in the spread of VBD with matrices M andM capturing vector-to-human and human-to-vector infections,respectively. Thus, in order to quantify indirect infectionsbetween humans mediated by vectors and vice versa, weshould iterate Eq. (13) obtaining(

�ε H

�ε M

)= β2λMHλHM

μMμH

(MM 0

0 MM

)(�ε H

�ε M

). (14)

From Eq. (14) it becomes clear that nontrivial solutions for�ε H correspond to the eigenvectors of matrix MM. Specifi-cally, given a metapopulation defined by �n, �m, R, and p, the

stationary solutions with infinitesimal incidence correspond toeigenvectors of MM whose eigenvalues can be written as

i = μMμH

β2λMHλHM. (15)

Under these conditions, the maximum eigenvalue max(MM)encodes the combination of the RM parameters that corre-sponds to the epidemic threshold, namely,

β2λMHλHM

μMμHmax = 1 . (16)

The former equation reveals the minimum infectivities, eitherλHM or λMH , that trigger the epidemic outbreak. To derive asimple critical infectivity one can set λMH = δλHM , so that

λMHc =

√μHμM

δβ2max(MM). (17)

To test the validity of Eq. (17) we have carried out ex-tensive numerical simulations in synthetic metapopulationsconsidering λMH = λHM = λ. The top panels in Fig. 3 showthe epidemic diagrams ρH (p, λ) by computing the fractions

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FIG. 3. Epidemic threshold and evolution of the leading eigenvector of matrix MM. (a)–(c) in the top show three epidemic diagramsρH (p, λ) in a synthetic metapopulation of N = 50 patches. Each panel corresponds to a different rescaling value (α) for the mobility ofinfected humans, namely: (a) α = 0, (b) α = 0.5 and (c) α = 1. In addition, we have set λHM = λMH = λ while the rest of the RM parametersare: μH = 0.3, μM = 0.3, and β = 1.0. The color code show the incidence ρH as obtained from agent-based simulations while the solidcurve represent the prediction for the epidemic threshold λc, calculated from Eq. (17). (d)–(f) show the evolution, as a function of p, of the Ncomponents of the eigenvector of matrix MM corresponding to maximum eigenvalue max(MM).

of humans infected ρH as a function of λ and p. From thesediagrams it becomes clear that, for each value of p, thereexists a critical value λc so that for λ > λc the epidemicphase appears. The border of this region [solid curves inFigs. 3(a)–3(c)] is the function λc(p) calculated with Eq. (17),showing an excellent agreement with the results obtained fromnumerical simulations.

V. ABRUPT TRANSITIONS OF LEADING PACTHES

Apart from the agreement between Eq. (17) and the nu-merical simulations, the evolution of the epidemic thresholdλc(p), reported in the three upper panels points out a nontrivialdependence with the degree of human mobility. Contrary towhat naively expected, human mobility can be detrimental toepidemics, as clearly illustrated in the panels for α = 0.5 and1.0. This counterintuitive effect of mobility was already foundfor SIR and SIS diseases in networked metapopulations [46]as a result of the redistribution of the effective populationsacross patches due to mobility. In the case of VBD, thisprocess corresponds to an homogeneization of the effectiveratios between vectors and humans so that a high risk patchwith large γi = mi/ni tends to decrease its effective value dueto the increase of neff

i caused by human mobility.A more striking phenomenon reported in the epidemic dia-

grams of Fig. 3 is revealed by the sharp variations in the slopeof the curves λc(p). These abrupt changes are the product ofcollisions between the two maximum eigenvalues of matrixMM as p varies. This way, the two maximum eigenvaluesinterchange their order at some critical mobility value pc.These collisions do not have an strong impact in the epidemic

threshold since the function λc(p) is continuous. However,they are the fingerprint of a sudden change in the form of theeigenvector corresponding to the maximum eigenvalue, �vmax,of matrix MM. This abrupt transition is of utmost importancesince the components of �vmax encode the most importantpatches driving the unfolding of the epidemics.

Let us recall that matrix MM incorporates the demographicinformation, �n, the mobility patterns, R and the vector distri-bution, �m, having as unique parameter the degree of mobilityp. Thus, for each value of p the spectral analysis of MMgives us the epidemic threshold λc and the distribution ofpatches triggering the epidemic onset in the components of�vmax. The evolution of the components of �vmax as a functionof p is shown in the bottom panels, (d–f), of Fig. 3. Fromthese plots, it becomes clear that the discontinuities of theslope of λc(p) correspond to abrupt changes in the formof �vmax. Namely, in the three cases, patch number 26 isthe one causing the epidemic onset for p = 0 and p � 1.This is obvious since patch 26 is the one with largest ratioγi = mi/ni in the synthetic metapopulation. However, as pincreases, the leading patch changes, being replaced by patch18 in the case of α = 0 (d) and α = 0.5 (e), while for α = 1(f) the leading patch is replaced by a collection of them.Remarkably, the case α = 0 shows a second abrupt transitionat p′

c 0.92.From a practical point of view, these abrupt changes point

out that containment strategies targeting a certain neighbor-hood can sharply change from efficient to useless due tosubtle changes in human mobility. In Appendix A, we confirmnumerically these abrupt changes in the suitability of targetedimmunization policies.

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VI. EPIDEMIC RISK OF GEOGRAPHICAL AREAS

Spurred by the ability of the Markovian formalism tocapture the dynamics of VBD and the insights provided bythe spectral properties of matrix MM for identifying the areastriggering the onset of epidemics, we move one step furtherand evaluate the epidemic risk associated with each patch.To this aim, we propose a theory-driven prevalence indicatorwhich serves as a proxy to determine the most exposed areasto the spread of VBD.

For this purpose, let us analyze the elements of the matricesM and M, defined in Eqs. (11) and (12). From these equationswe realize that the elements Mi j (Mi j) contain all the possi-ble microscopic contagion processes from vectors (humans)associated with patch j to humans (vectors) associated withi. Therefore it is possible to estimate the effective numberof human-human interactions mediated by vectors that anindividual from subpopulation i receives from those with res-idence in j. This quantity, denoted as Ci j , can be obtained as

Ci j =N∑

k=1

MikMk j . (18)

After introducing the expressions of Mj and Mk j fromEqs. (11) and (12) and after some algebra, the explicit expres-sion for the indirect human-human contacts reads

Ci j = δi j (1 − αp)(1 − p)mini(neff

i

)2

+αp(1 − p)Rjimin j(neff

i

)2

+ (1 − αp)pRi jmjn j(neff

j

)2

+ αp2∑

k

RikR jkmkn j(neff

k

)2 . (19)

This expression takes into account all the possible infectionpathways connecting humans in patch j to an individual withresidence in node i. In particular, those infections may takeplace in four possible ways [each one encoded by the termsof Eq. (19)]: (i) when an infected individual from patch jremains at his/her residence, transmits the disease to onevector in j which then transmits the disease to another residentin j; (ii) when an infected individual from patch j visits patch iand infects a vector that will later pass the disease to a residentof patch i; (iii) when a resident of patch j infects a vector thereand a healthy human traveling to from i to j gets infected; and(iv) when both the infected individual from j and the healthyindividual from i travel to a contiguous third patch k wherethe infection takes place mediated by a vector.

Finally, in order to make predictions about the impact ofthe VBD on a geographical area, i, we must account for allthe possible infections from each patch of the metapopulationand to weight the resulting number by the population of i.This way, the epidemic risk indicator for each patch i, in thefollowing denoted as ERi, can be defined as

ERi = ni

N∑j=1

Ci j = ni

N∑j=1

(MM)i j . (20)

The evaluation of ERi, as defined above, can be done directlywithout the need of neither numerical simulations nor makingthe integration of the Markovian equations. In fact, once dataabout demography, vector distribution and mobility patternsare available, one can estimate the epidemic risk of eachsubpopulation.

To validate the Epidemic Risk measure, we now move to areal metapopulation, the city of Santiago de Cali (Colombia).With a population of more than 2 millions of inhabitants,it offers the possibility of comparing our predictions in ascenario for which severe epidemic outbreaks of VBD arerecurrently found. In particular, due to its location and cli-mate, Cali is a Dengue endemic city in which records of thehistorical incidence of this disease are available for compar-ison. To this aim, we collected demographic and mobilitydatasets [68] whereas vectors abundance across districts wasobtained from entomological reports made yearly by the localauthorities [69] (further information in Appendix B). With thisinformation at hand, and by using Eq. (20), we assign theepidemic risk of each of the 22 districts in which the city ofCali is divided. These epidemic risk values are compared tothe observed Dengue incidence across the 22 patches duringthe years 2015 and 2016 [70,71].

In Fig. 4, we show this comparison by normalizing thevalues of both epidemic risk and Dengue incidence by theirmaximum observed value (in both cases that of district 13). Inparticular, we find a coefficient of determination of R2 = 0.81,indicating that the proposed prevalence indicator is able tocapture the spatial distribution of Dengue cases across thecity. On more general grounds, this agreement points out thatgiven the demography, the commuting patterns and the spatialdistribution of vectors across a given population, one can useEq. (20) to identify areas where containment measures shouldbe promoted to reduce the impact of possible outbreaks.

VII. CONCLUSIONS

The control of infectious diseases represents one of themajor societal challenges. Understanding the complex inter-dependency between human activity and contagion processesis key to explain the onset and development of large-scaleepidemics. Here, focusing on VBD, we have integrated in-formation from urban daily commutes and the geographicdistribution of humans and vectors to estimate the epidemicrisk associated with different connected regions. In particular,we have provided a metapopulation formalism to assess therole that the former ingredients play on the propagation ofVBD. We have proved that this formalism constitutes a veryreliable and time-saving platform, since its Markovian equa-tions enable to reproduce very accurately not only the globalincidence of VBD but also the spatiotemporal spreading pat-terns observed in Monte Carlo simulations.

Based on this agreement, we have derived an analyticalexpression of the epidemic threshold that captures the criticalconditions which leads to the onset of epidemics. Apart fromthe detrimental effect that mobility may have on the spreadof diseases, the study of the epidemic threshold has revealedinteresting phenomena such as the existence of abrupt changesin the way epidemics unfold. In particular, we have shownthat the subset of patches leading the epidemic onset can

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FIG. 4. Real Dengue incidence vs estimated epidemic risk inthe city of Cali (Colombia). (Top) Normalized epidemic risk(ER/ERmax) vs normalized Dengue incidence (I/Imax) for each ofthe 22 districts of Cali. Color encodes the Epidemic Risk, fromthe lower (yellow) to the highest (blue).The correlation between thetwo variables yields a coefficient of determination of R2 = 0.81.(Bottom) Spatial distributions of the normalized Dengue incidencein the city of Cali (left) and the normalized epidemic risk (right)according to Eq. (20). The parameters concerning agent mobilityhave been set to (p, α) = (0.36, 0.75).

suddenly change as human mobility varies. This phenomenonhighlights the need of incorporating real human mobility pat-terns into the design of containment policies targeting specificgeographical areas, for efficient policies can turn useless dueto a small variation of human mobility habits.

Finally, relying on the matrix containing the informationabout the effective number of human-human contagions, wehave derived an epidemic risk indicator that allows us toclassify the patches according to their exposure to VBD.By computing this epidemic indicator, we have reproducedwith great accuracy the geographical distribution of Dengueincidence in the city of Santiago de Cali (Colombia), whereDengue in an endemic disease, thus being able to identify themost vulnerable areas where prevention measures should bepromoted.

In a nutshell, our results point out that the spread ofVBD is the result of a delicate interplay between commutingflows, human census and vector distribution. This interplayis captured both in the analytical expression of the epidemicthreshold and in the epidemic risk indicator. As a result, we

have shown that small variations of the former ingredients,such as the degree of mobility, can lead to abrupt changesin the way epidemics unfold. Our framework, although con-taining several simplifying assumptions to allow the analyticaltreatment, has shown useful to integrate human and contagiondynamics and it can be readily implemented to identify thoseregions where immunization policies should be reinforced andto forecast the consequences of control strategies focused onmobility restrictions.

As future work, the formalism here presented paves theway to the incorporation of additional features that, togetherwith human mobility, are key for the dissemination of vector-borne diseases. Among these features we find (a) the inclusionof seasonality modulating both, the population and the bittingrate of vectors [72] or the effects of climate change on vectorscensus [6–9], both ingredients can be incorporated as anexternal driver for each patch through the use of the multiplexformalism [73–75]; (b) the time-varying nature of humancontact patterns [76–79] and the adaptability of mobility be-haviors in epidemic scenarios [80,81]; and (c) the possibilityof incorporating more refined compartmental models withthe addition of multiple states [82,83] to address the impactthat direct human-human transmission in some VBD suchas Zika has on the epidemic threshold [62], or the interplayof several VBD with competitive or cooperative interactions[84–86]. These ingredients can be readily incorporated intothe Markovian model, improving its capacity to evaluate therisk of specific VBD in particular regions.

ACKNOWLEDGMENTS

We are especially grateful to A. Arenas and O. Vasilievafor useful comments and discussions. This work was partiallysupported by Universidad del Valle (grant CI-165), Gobiernode Aragon/Fondo Social Europeo (Grant No. E36-17R), Min-isterio de Economia, Industria y Competitividad (MINECO)and Fondo Europeo de Desarrollo Regional (FEDER) (GrantsNo. FIS2015-71582-C2 and No. FIS2017-87519-P) and bythe Spanish Ministry of Science, Innovation and Universities,the National Agency for Research Funding AEI and FEDER(EU) under the grant PACSS (RTI2018-093732-B-C22) andthe Maria de Maeztu program for Units of Excellence in R&D(MDM-2017-0711).

APPENDIX A: EFFICIENCY OF TARGETEDIMMUNIZATION POLICIES

The study of the critical properties of VBD in the main texthas revealed the existence of some mobility values, denotedas pc, for which the components of the leading eigenvec-tor change abruptly. Translated into epidemiological words,this phenomenon pinpoints the change in the most affectedpatches, which is of great relevance since targeted policiesin specific areas can pass from useful to useless as humanmobility varies.

To prove it, we now study the effects of applying preven-tion measures in specific locations selected according to thelargest components of the leading eigenvector of the criticalmatrix MM. We consider the synthetic network used in themain text and set α = 0.5, for which the change in the leadingpatch happens at p = pc = 0.38 (see Fig. 3). To study the

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FIG. 5. Temporal evolution, according to the Markovian equations, of the number of infected agents by a VBD which spreads over a BAnetwork. The black line denotes the original curve where no policy has been implemented whereas dashed lines correspond to the case in whichpatches 26 (red) and 18 (blue) are immunized. The epidemic parameters have been set to (α, λHM , λMH , μH , μM ) = (0.5, 0.25, 0.25, 0.3, 0.3).The values for human mobility are (a) p = 0.1 (p < pc) and (b) p = 0.5 (p > pc).

effects of containment measures, we assume that immunizedagents are no longer susceptible of contracting the disease,so λMH = 0 for them. Finally, we target two different sub-populations for the immunization policies: patch 26 that isthe leading patch for p < pc, and patch 18 that sustains theepidemic outbreak when p > pc.

Figure 5 confirms that the effectiveness of targeted policiesagainst VBD is strongly influenced by aspects concerning hu-man mobility. In fact, immunizing agents from patch 18 leadsto the extinction of the disease for p = 0.5 > pc, whereas itis almost ineffective for mobility values below this threshold(p = 0.1 < pc). This result, along with the others presentedin the main text, highlights the importance of designing con-tainment policies taking into account the complex interplaybetween human mobility patterns, census data and vectorabundance.

APPENDIX B: DESCRIPTION OF CALI DATASET

One of the most important contributions of this work is theformulation of a new framework which can easily incorporatemobility data of real cities to address real epidemic scenarios.In the main text, we tackle the spread of VBD in the cityof Cali (Colombia), whose geographical and meteorologicalfeatures make it an endemic region for several VBD such asDengue, Chikungunya or Zika. In particular, here we focus onthe spread of Dengue.

To assess the effect of mobility on the spread of Denguein Cali, it is necessary to reconstruct the mobility networkof its inhabitants from data. For this purpose, we dividethe city into 22 districts, which correspond to the officialadministrative divisions called comunas. Regarding demog-

raphy, the population distribution across comunas has beenextracted from census data that the municipality facilitates[68]. Mobility flows connecting comunas are extracted fromurban commuting surveys [87]. As a result, more than 105

trajectories were recorded, which suppose a representativesample of Cali’s commuting flows. Once all the data havebeen gathered, an origin-destination matrix, encoded in ourformalism by matrix R, is computed as

Ri j = Wi j∑Nl=1 Wil

, (B1)

where the numerator corresponds to the number of tripsbetween patches i and j while the denominator countsall the reported trips departing from patch i. The resultis a weighted directed network encoding the probabilitythat an agent visits other neighborhoods different from itsresidence.

Apart from the mobility network, the distribution of vectorsacross the city also plays a crucial role in the outcome ofthe disease. The number of vectors inside a geographicalregion is strongly linked to environmental features such as al-titude, temperature and humidity but also to human-dependentfactors like health and economic conditions. To model vec-tor distribution across comunas, we use as a proxy the so-called recipient index. This quantity encodes the probabilityof finding vector pupae in different recipients which havebeen previously distributed across the city. A high value ofthe index means a higher probability of finding vectors. Forthis reason, we assume that the ratio between the number ofvectors and humans inside each patch in our model is directlyproportional to its recipient index, which is extracted fromentomological data of the year 2015 [69].

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